A moyo is essentially an area of the board that is more or less loosely enclosed by stones of the same color. It is not to be confused with actual territory. To say something is territory in Go is to say that it will still be territory at the end of the game. A moyo is generally so large that if it were actually territory, its owner would win the game. Therefore, the opponent must invade or reduce the moyo before its owner has a chance to solidify it. In the process of removing chunks of the moyo, the opponent allows the owner the opportunity to solidify part of the moyo into actual territory. Thus, moyo strategy may be more or less summed up (for laypeople) as follows: Instead of enclosing n points of territory, try to enclose 3n points not so strongly as to be territory (as that would be impossible), but firmly enough that when attacked, something like half of it will turn into actual territory, which is more than you would have gotten if you'd gone for territory right off the bat. To clarify a little bit, here's a full-goban diagram that might help (note that this situation would be unlikely to arise in an actual game, but it's useful to illustrate):

(The comma is a hoshi (star point) and is for reference only. People who don't know what it is can ignore it, as it has no game effect.)

The white (o) stones enclose territory in the corners. The points therein are pretty much unassailable. Black (x), on the other hand, has no one point that is guaranteed to be territory, despite having 10 stones on the right side of the board. However, the idea is that although no one point is guaranteed to be territory, it is pretty much guaranteed that at the end of the game, a large percentage of the right side of the board will belong to black at the end of the game, because in invading one section, white will allow black to strengthen another section.

So, what do you do when your opponent builds a moyo? Well, the worst thing you can do is ignore it, since he'll either just continue to expand it, or start fortifying and turning it into territory. You basically have three options.

Invade. This is the most aggressive strategy, and needs the most skill to make work. Invading means playing a stone fairly deep inside the moyo, then trying to either connect out to a living group of your own, or form two eyes and live on the spot. Both can be quite tricky. Generally, one must make sabaki, which is quite tricky. This means making a very light formation of stones in the opponent's moyo, so that the opponent can capture some of them, but must allow some of the others to escape in the process. This is the domain of strong players, and I'm not qualified to talk much about it.

Reduce. Less aggressive than invasion, reduction involves playing stones just a little bit inside the moyo, so that the opponent is more likely to play further inside to push your stones out, rather than playing outside your stones and forcing you to invade. In the process, you've succeeded in squeezing the walls of his moyo slightly inwards, so that the enclosed territory is less. There is a proverb that "Reduction is worth as much as invasion." It's usually much safer, too.

Build your own moyo. Aggressive in a different way, it's also often feasible to try to "out-moyo" your opponent. One must be careful though. It is often said that in a moyo vs. moyo game, you should check to see whose moyo is bigger. If your opponent's is bigger, try to reduce it (invasion might have the unintended effect of pushing him out into your moyo, which would be disatrous), and if yours is bigger, just keep enlarging it until he reduces or invades. It is generally not recommended to try to "catch up" if your moyo is smaller, since you'll end up overplaying and stretching yourself too thin in the attempt to expand faster. Then your opponent will attack and destroy everything you built.

Moyos can be a useful concept to apply to real life, like many concepts in Go. The real life interpretation is this: instead of trying to make sure you gain a specific advantage out of a situation, try to set the situation up so that no matter what happens, you get something out of it. For instance, if trying to get a girlfriend, it might be a better idea to try to put yourself in social situations where it's easy to meet a large number of potential girlfriends, so that your odds are good of ending up with one of them, than to pick one specific girl and devote all your energies to trying to get her. I'm sure there are plenty of other examples, but this is the one that sprung to mind.